Hurdle negative binomial distribution

Probability mass function, distribution function, and random generation for the hurdle (zero-altered) negative binomial distribution.

Usage

dhnbinom(x, size, prob, zeroprob = 0.5, log = FALSE)

phnbinom(q, size, prob, zeroprob = 0.5, lower.tail = TRUE, log.p = FALSE)

rhnbinom(n, size, prob, zeroprob = 0.5)

Arguments

x, q

integer vector of counts

size

dispersion parameter, must be strictly positive

prob

probability of success in each trial, in (0,1)

zeroprob

probability of a zero, between 0 and 1

log, log.p

logical; return log-density if TRUE

lower.tail

logical; if TRUE, probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).

n

number of random values to return.

Value

dhnbinom gives the probability mass function, phnbinom gives the distribution function, and rhnbinom generates random deviates.

Details

This implementation allows for automatic differentiation with RTMB.

A hurdle distribution models the zeros and the positive counts as two separate processes: the probability of a zero is a free parameter, and the positive counts follow the corresponding zero-truncated distribution. Writing \(p_0\) for zeroprob, $$P(X = 0) = p_0, \qquad P(X = x) = (1 - p_0)\,\frac{P_{\mathrm{NB}}(x;\,r,\pi)}{1 - \pi_0}, \quad x = 1, 2, \ldots$$ where \(\pi_0 = P_{\mathrm{NB}}(0;\,r,\pi)\) is the probability of a zero under the ordinary negative binomial.

Unlike zero-inflation, which can only add zeros to those the negative binomial already produces, zeroprob here is exactly the probability of a zero and may be larger or smaller than \(\pi_0\). The two coincide with the ordinary negative binomial when zeroprob equals \(\pi_0\).

References

Mullahy, J. (1986) Specification and testing of some modified count data models. Journal of Econometrics, 33, 341-365.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC, doi:10.1201/9780429298547. An older version can be found in https://www.gamlss.com/.

See also

hnbinom2, hpois, hbinom, zinbinom, ztnbinom

Examples

set.seed(123)
x <- rhnbinom(5, size = 2, prob = 0.4, zeroprob = 0.3)
d <- dhnbinom(x, size = 2, prob = 0.4, zeroprob = 0.3)
p <- phnbinom(x, size = 2, prob = 0.4, zeroprob = 0.3)